Bipartite theory on Neighbourhood Dominating and Global Dominating sets of a graph

Bipartite theory of graphs was formulated by Stephen Hedetniemi and Renu Laskar in which concepts in graph theory have equivalent formulations as concepts for bipartite graphs. We give the bipartite version of Neighbourhood sets, Line Neighbourhood sets and global dominating sets.


Introduction
By a graph G = (V, E) we mean a finite, undirected graph with neither loops nor multiple edges.The order and size of G are denoted by p and q respectively.For graph theoretic terminology we refer to Harary [1].
Let G = (V, E) be a graph.Let v ∈ V .The open neighbourhood and the closed neighbourhood of v are denoted by N (v) and N [v] = N (v) ∪ {v} respectively.If S ⊆ V , then N (S) = ∪ v∈S N (v) and N [S] = N (S) ∪ S. Let S ⊆ V .We denote the subgraph induced by the set of vertices S as S .
A subset S of V is called a dominating set of G if N [S] = V .The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G).For further results on domination the reader is referred to an excellent book on fundamentals of domination [2] and a survey of advanced topics in domination [3].
Various types of domination have been defined and studied by several authors.E.Sampathkumar introduced the concept of neighbourhood set [7], line neighbourhood set [8] and global dominating set [9] . The minimum cardinality of a line neighbourhood set is called the line neighbourhood number of G and is denoted by n 1 0 (G).A dominating set D ⊆ V is called a global dominating set of G if D is a dominating set in both G and G.The minimum cardinality of a global dominating set is called the global domination number of G and is denoted by γ g (G).
For any problem, say P , on an arbitrary graph G, there is a corresponding problem Q on a bipartite graph G * , such that a solution for Q provides a solution for P .The so called bipartite theory of graphs was introduced by Hedetniemi and Laskar in [5,6].Let G = (X, Y, E) be a bipartite graph.Two vertices u, v ∈ X are X−adjacent if they are adjacent to a common vertex in Y .By the X−neighbourhood of a vertex u of X, we mean the set N X (u) = {v ∈ X : u and v are adjacent}.Let S ⊆ X and Bipartite theory on domination in complement of a graph, irredundant set of a graph and dominator coloring of a graph are studied in [10,11,12,13].Here we give the bipartite theory of neighbourhood set, line neighbourhood set and global dominating set of a graph.

Bipartite Constructions
Given an arbitrary graph G, we can construct bipartite graphs G 1 = (X, Y, E 1 ) which represents G, in the sense that given two graphs G and H, G is isomorphic to H if and only if the corresponding bipartite graphs G 1 and H 1 are isomorphic.
We give below the bipartite constructions of V E(G), EV (G) as given in [5] and super duplicate graph D * (G) as defined in [4].
The cardinality of a smallest X-neighbourhood set of G is called the X-neighbourhood number of G and is denoted by n X (G).

Consider the graph
The set Proposition 2.2.In a bipartite graph G, every X-neighbourhood set is a Xdominating set.
The converse of the proposition 2.2 need not be true.Consider the graph given below, S = {b} is a X-dominating set.Remark 2.6.
The converse of the proposition 2.3 need not be true.Consider the graph given below.The set S = {u 2 } is a X-neighbourhood set.
is the subgraph of G induced by v and all vertices adjacent to v. The neighbourhood number n 0 (G) of G is the minimum cardinality of a neighbourhood set of a graph G.For a line x = uv in 176 Y.B. Venkatakrishnan and V. Swaminathan where S(G) denotes the subdivision graph of G namely the graph obtained from G by subdividing each edge of G exactly once.The bipartite graph EV (G): The graph EV (G) = (E, V, J) is defined by the edges J = {(e, u)(e, v)/e = (u, v) ∈ E}.Super Duplicate graph D * (G): The bipartite graph D *

3 . 3 . 1 .
X − B is hyper independent set.✷ Bipartite theory of X−neighbourhood set Theorem For any graph G, n 0 (G) = n X (V E(G)).Proof: Let S be a n 0 (G)− set.G = x∈s N X [x] .N [S] = V and every edge of G is induced by N [S] of G.In graph V E(G) = (X, Y, E 1 ), N X [S] = X.Let e = uv ∈ E(G).Then u, v ∈ X and e ∈ Y .If e is adjacent to u and u ∈ S, then e will be in V 1 .If u / ∈ S and e is adjacent to v. If v ∈ S then e ∈ V 1 otherwise, e is adjacent to vertices in N Y (x).e ∈ V 1 .Therefore, S is a X-neighbourhood set in V E(G).Therefore, n X (V E(G)) ≤ |S| = n 0 (G).
is not a Y -dominating set.Since, u 6 is not adjacent to u 2 .
[11]rem 2.7.[11]In a bipartite graph G, A subset D of X is a Y −dominating set if and only if X − D is a hyper independent set.Proposition 2.8.For any bipartite graph G without isolated points, n X (G) = γ Y (G) if and only if there exists a minimum n X − set S such that X − S is hyper independent set.Proof: Let X − S be hyper independent set.S is a Y -dominating set.Hence, γ Y